What does an integral tell you?
I have been losing my mind over this for a good bit, so bear with me. I've been looking into generalizing the ideas of calculus into more dimensions, and for differentiation, I've found it to be...
View ArticleAbout spaces of the second category
I have a presentation in which I am asked to demonstrate the following result:A metric space $(X, d)$ is of the second category if and only if any sequence of non-empty open dense sets in $X$ has a...
View ArticleHow to prove the Cauchy Criterion for Product?
I am trying to figure out how to prove the Cauchy Criterion for products apply for all $\epsilon$.Specifically {$u_n$}^$\infty$_${n=1}$ is a sequence of positive real numbers such that for all...
View ArticleCan you explain me integral approximation for finite sum...
Can you please explain me the integral approximation for a finite sum in easy language for the question $$\lim\limits_{n\to\infty}\int_{k=n}^{2n}\frac{1}{2k+1}$$
View ArticleQuestion about a proof
Corollary 7.9. Let a function $f: \mathbb{R} \rightarrow \mathbb{C}$ be integrable with respect to a Lebesgue-Stieltjes measure $\mu$ . Then for any $\varepsilon>0$, there is a continuous function g...
View ArticleWhat does existence of the Real numbers mean?
It is a common practice in real analysis text book to show that a complete ordered field exist, this ordered field is then called the Real numbers. What does this existence mean (where does it exist)...
View ArticleAn averaging inequality for continuous functions [closed]
Let $a\in (0,1)$ and $f$ be a continuous and integrable function on $[0,\infty)$. Is it true that$$\left|\int_{0}^{1} (1-t)^{-a}{f(x t)}dt \right|\geq C |f(x)|$$for all $x>0$, for some constant $C$...
View ArticleIs there any Banach space other than l^p satisfies these conditions?
Let $X$ be a Banach space.There exists a biorthogonal system $(x_i, f_i)_{i \in \mathbb{N}},$ where $(x_i)_{i \in \mathbb{N}} \subset X$ and $(f_i)_{i \in \mathbb{N}} \subset X^*$ such that for some...
View ArticleHow many PDEs does it take for a system to be unsolvable in general?
Given a system of PDEs with $n_f$ functions $f_i(x_1, x_2 ... n_v)$ each of $n_v$ real variables how many equations $n_c$ does it take before the system is unsolvable in general? I am guessing that as...
View Article(Reference request) Asymptotic of generalized exponential integral
I'm interested in asymptotics of the integral$$f_a(x)=\int_1^{x} t^a e^t\,dt, \ a\in \mathbb R,$$as $x\to+\infty$.Mathematica gives $f_a(x)\sim x^a e^{x}$, $x\to \infty$.I didnt' find it in...
View Articlerelationship between boundedness, and uniform convergence for functional serise.
If a functional series is not bounded on I, and converges to some function f, that doesn't imply that f_n is not uniformally converges, yet i don't know wheremy teacher have deduced this result, check...
View ArticleL^1 function with zero average on balls of fixed radius
Let $f \in L^1(\mathbb{R}^d)$ be a function such that\begin{align*}\int_{B_1(x_0)} f(x)\,dx = 0\end{align*}for every $x_0 \in \mathbb{R}^d$ (where $B_1$ denotes the unit ball). I want to prove that $f...
View ArticleIntegral $\iint \limits_{{x,y \ \in \ [0,1]}}...
Hi I am trying to integrate $$\mathcal{I}:=\iint \limits_{{x,y \ \in \ [0,1]}} \frac{\log(1-x)\log(1-y)}{1-xy}dx\,dy=\int_0^1\int_0^1 \frac{\log(1-x)\log(1-y)}{1-xy}dx \,dy$$A closed form does exist. I...
View ArticleA question about linear change of variables
enter image description here\begin{document}\noindent (ii) \textbf{(Linear change of variables)} Let $T$ be a $n \times n$ invertible matrix with real entries. Then[\int f(T(x)) |\det T| , dx = \int...
View ArticleProve that there are uncountably many sets Si ⊆ℕ such that, for any Si , Sj ,...
My first try was using Cantor style diagonal argument. Let's take Si ⊆ℕ be an infinite subset of natural number. Then Si can be represented as a binary sequence. Let's take a collection of subset in a...
View ArticleProving the convegence of a alternating sequence
Prove that the series given below is convergent.$$\sum_{n=1}^{\infty}(-1)^n\frac{(n+1)^n}{n^{n+1}}$$The original problem was to investigate the convergence of the series $\displaystyle...
View ArticleShow convergence of integral $\int\limits_1^\infty\frac{1}{P(x)}dx$
Let be $P(x)$ a polynomial of degree $n$,i.e. $P(x):=a_0+a_1x+\dotsc+a_nx^n$. We assume that all zero spots of $P(x)$ are negative. Show that $\int\limits_1^\infty\frac{1}{P(x)}dx$ converges absolutely...
View ArticleNumerical verification for a $2D$ Euler-Mascheroni Constant $\gamma$
Definitions$\gamma$ is the classical Euler-Mascheroni constant.$\zeta(s)$ is the Riemann zeta function.$\Gamma(s)$ is the Gamma function.$\delta_{n,1}$ is the Kronecker delta, equal to $1$ when $n=1$...
View ArticleDecide for which values of $p \in [1, +\infty]$ the following statement holds.
Problem: Let $L^p(\mu)$ be the space of equivalence classes of functions $f$ defined on the unit circle and measurable with respect to the Lebesgue measure $\mu$, such that$$\|f\|_p =...
View ArticlePlease help me with following limit related to Stolz–Cesàro theorem
If $x_1\in(0,1)$ and $x_{n+1}=x_n(1-x_n)$, find $\lim_{n\rightarrow\infty}nx_n$ (Hint: using Stolz–Cesàro theorem)I can prove that $x_n$ monotonically decrease and...
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